Prediction of phonon-mediated superconductivity in new Ti-based M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_2$$\end{document}2AX phases

A high-throughput computational method is used to predict 39 new superconductors in the Ti-based M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_2$$\end{document}2AX phases, and the best candidates are then studied in more detail using density functional theory electron–phonon coupling calculations. The detailed calculations agree with the simple predictions, and Ti\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_2$$\end{document}2AlX (X: B, C and N) materials are predicted to have higher values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_c$$\end{document}Tc than any currently known hexagonal M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_2$$\end{document}2AX phases. The electronic states at the Fermi level are dominated by the Ti 3d states. The choice of X (X: B, C and N) has a significant impact on the electronic density of states but not on the phonon characteristics. The electron–phonon coupling parameter for Ti\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_2$$\end{document}2AlX (X: B, C and N) was determined to be 0.685, 0.743 and 0.775 with a predicted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_c$$\end{document}Tc of 7.8 K, 10.8 K and 13.0 K, respectively.

www.nature.com/scientificreports/ As our high-throughput model predicted the highest T c for Ti 2 AlX (X: B, C and N), we therefore performed a more detailed study of superconductivity in these materials Whilst the structural, electronic, elastic, thermodynamic, and vibrational properties of Ti 2 AlX (X: B, C and N) have been studied theoretically [33][34][35][36][37] and experimentally 35,38 , we are unaware of any study of superconductivity in these materials. Thus, this work presents an ab initio study of the superconducting T c , including electron-phonon coupling, as well as structural, electronic and phonon properties of Ti 2 AlX (X: B, C, and N), using the plane-wave pseudopotential approach to density functional theory (DFT). The Eliashberg spectral function is calculated by combining linear response theory 39,40 with Migdal-Eliashberg theory 41,42 . These quantities are then used to investigate the origin of superconductivity in these materials, and the effect of changing X (X: B, C, and N). In our previous study of superconductivity in Nb-C based M 2 AX phases 21 we found that the Migdal-Eliashberg predictions agreed with experimental T c values within ± 1 K.

Methods
The calculations used the Quantum Espresso ab initio simulation package 39,40,43 with the Perdew-Burke-Ernzerhof (PBE) 44 exchange-correlation approximation and ultrasoft pseudopotentials 45 . The plane-wave basis cut-off is 60 Ry ( ∼ 812 eV) and the Brillouin zone integration used the Monkorst-Pack 46  Phonon calculations used the linear response approach 39,40,43 and the Brillouin zone integration for the phonons used a ( 4 × 4 × 4 ) q-mesh and twelve dynamical matrices by symmetry. The electron and phonon results are combined to compute the electron-phonon interaction using the Migdal-Eliashberg theory 41,42 and hence T c .
This calculation of T c is very computationally demanding, and is therefore impractical for high-throughput screening for novel superconductors. A change in the A element in Ti 2 AX (X: B, C, and N) appears to have a comparable impact to the superconducting isotope effect, and in our previous work 21 on Nb-C based M 2 AX phases, we showed that a simple model based upon the Fröhlich 47 theory of the isotope effect was an effective foundation for a high-throughput screening approach of these materials, with where M is the mass of a formula unit, N(E F ) is the electronic density at the Fermi energy E F and T 0 and α are linear fit parameters. This model has a critical value of N(E F )/ √ M > T 0 /α for superconductivity to occur. This functional form is an approximation to the simplified BCS equation 48 .
In its most basic form, BCS theory gives the superconducting transition temperature T c in terms of the electron-phonon interaction (V) and the Debye temperature ( D ), and can be simplified as This exponential form is approximately linear when 0.2 < NV < 0.7 and saturates at large values of NV. Within each Ti 2 AX family (borides, carbides and nitrides) we might expect a similar V and hence observe behaviour similar to Eq. (1).
The advantage of Eq. (1) is that N(E F ) can be calculated in much less time (typically less than 1 core hour) than the electron-phonon matrix elements (typically 300 core hours per material), and when combined with the observed trend in T c vs. N(E F )/ √ M for known superconducting materials (Ti 2 GeC, Ti 2 InC and Ti 2 InN), it can be used in a high-throughput search to predict the superconducting transition temperatures of candidate materials for which there have been no previous superconductivity studies. The most promising of these is Ti 2 AlX(X: B, C and N) which is then investigated in more detail using full electron-phonon coupling and Migdal-Eliashberg theory.

Results
Superconducting T c results. The high-throughput screening is based on our Fröhlich model, which predicts a linear relationship between the critical temperature ( T c ) and the value of N(E F )/ √ M (as shown in Fig. 1). As shown in the Supplementary Information, the T c values of three known superconductors (Ti 2 GeC, Ti 2 InC and Ti 2 InN) were calculated using the Eliashberg theory 41,42 and the best fit to experimental T c values is found when µ * = 0.13 for all 3 materials. These theoretical values are shown in Fig. 1 using blue and magenta squares, whilst the corresponding experimental T c values are given in black circles.
In our previous work 21 we found that Nb-C based M 2 AX phases containing Al had high T c values. Hence we used the Eliashberg theory with µ * = 0.13 to determine the superconductivity temperatures of Ti 2 AlX(X: C and N) and Ti 2 GeN materials, none of which have a known T c . These results are also plotted in Fig. 1 as blue and magenta squares, and it appears that the M 2 AX carbides and nitrides fall into 2 distinct classes, which we have shown by 2 straight lines.
Recently, a new family of boride M 2 AX phases 49 has been synthesized but has yet to be tested for superconductivity. We therefore used Eliashberg theory to calculate the superconductivity temperatures of Ti 2 AB(A: Al, Ge, In) using µ * = 0.13 as before, for a direct comparison with the carbides and nitrides. These results are also plotted in Fig. 1 as red squares, and it appears that the borides fall into a third distinct class.
With electron-phonon coupling calculation results of T c for 3 materials in each of the Ti-based boride/ carbide/nitride families, we can perform a simple linear fit and use this as the basis for our Fröhlich model. The www.nature.com/scientificreports/ results are summarized in Table 1, and validate the usefulness of our screening method to predict T c in novel materials.
We can now use this model in a high-throughput screening approach to quickly evaluate similar materials for which T c is unknown. This only requires an electronic DOS calculation ( ∼ 1 core hour/material) which is much quicker than calculating the full electron-phonon coupling ( ∼ 300 core hours/material).
The full set of Fröhlich model results, for Ti 2 AX (where A: Al, Si, P, S, Cu, Zn, Ga, Ge, As, Cd, In, Sn, Tl and Pb; X: B, C and N) are shown in Table 2. This shows that the compounds containing Al have the highest superconducting transition temperature for each family, and that Ti 2 AlN is predicted to have the highest superconductivity temperature of any known M 2 AX material. Similarly, Ti 2 AlC is predicted to have the highest T c of the carbides, and Ti 2 AlB is predicted to have the highest T c of the borides.
As a validation of Eq. (1), we calculated values for D using Quantum Espresso's QHA package, and these are shown in Table 3 along with the values for T c and N(E F ) . This allows the electron phonon interaction strength (V) to be inferred from Eq. (2). This shows that the borides/carbides/nitrides all follow the same universal form as shown in Fig. 2.
Having presented the T c results, we now consider the detailed electronic and phonon properties that underly this behaviour.
Structural, electronic and Fermi surface properties. Ti 2 AlX (X: B, C and N) crystallizes in the hexagonal structure with space group P6 3 /mmc. The primitive unit cell has two formula units (eight atoms), each of which has occupied Wyckoff coordinates 4f (1/3, 2/3, z) for Ti, 2d (1/3, 2/3, 3/4) for Al, and 2a (0, 0, 0) for B (C and N) atoms. Thus, two lattice parameters, a and c, and one internal structural parameter, z, determine the structure. The hexagonal unit cell is shown in Fig. 3a in which blocks of edge-sharing Ti 6 X octahedra are sandwiched between planes of Al. The hexagonal Brillouin zone is shown in Fig. 3b.    ,c), the optimum internal parameter (z) and the bulk modulus (B) for each compound. These are shown in Table 4 and are in excellent agreement with previous theoretical results 33,34,36,37 . Figure 4 shows the electronic properties of hexagonal Ti 2 AlX including the band structure in the Brillouin zone, the total and partial density of states (DOS and PDOS), and the Fermi surface. The electronic density of states at the Fermi level (N(E F )) is important for metallic phases and superconductivity calculations. The PDOS of each component, broken into site and angular momentum contributions, is shown in Fig. 4, and as bands (mostly Ti 3d states) cross the Fermi level, this is the origin of the metallic behaviour. The electrical band structure is similar to that observed in previous studies 35,37 . Notably, each material has six valence bands that cross the Fermi level.
An analysis of the PDOS of Ti 2 AlN reveals that the lowest energy region, − 16.8 < E < − 15.6 eV, is dominated by the N 2s states with minor contributions from Ti 4s and 3d states. In Ti 2 AlC this low-energy region is     www.nature.com/scientificreports/ dominated by the C 2s orbital and is about 5 eV higher than in Ti 2 AlN. This low-energy region is absent in Ti 2 AlB material. The main valence band region, − 8.0 < E < − 4.0 eV, is made of hybridised Ti 3d and X 2p states, indicating covalent Ti-X bonding, with increasing covalent character in the order of B > C > N . Hence it is expected that Ti 2 AlN should have higher electrical conductivity than Ti 2 AlB and Ti 2 AlC. The region − 4.0 < E < E F eV is dominated by Ti 3d states with some weak Ti 3d and Al 3p hybridization. In the region E > E F the band structure is almost entirely dominated by Ti 3d states.  www.nature.com/scientificreports/ Qualitatively, it appears that the band structures around the Fermi level for Ti 2 AlX (X = B, C or N) are similar, with the principal difference being a shift of E F by 0.5 eV from X = B to X = C, and by 0.2 eV from X = C to X = N. This shift results in a significant increase in N(E F ). The electronic states at the Fermi level are crucial for superconductivity and we find that the DOS of Ti 2 AlN at the Fermi level is N(E F ) = 4.568 states/eV, with about 89.3%, 5.9% and 4.8% contributions from Ti, Al, and N atoms, respectively. As a consequence, the conduction properties are dominated by the Ti 3d electrons. Similarly, N(E F )= 3.048 states/eV for Ti 2 AlC and 2.241 states/ eV for Ti 2 AlB. This suggests that the most important contribution to the formation of the superconducting properties of Ti 2 AlX phases come from the Ti 3d states which increase N(E F ) and enhance according to the McMillan-Hopfield expression 50 : where ω 2 denotes the average squared phonon frequency, I 2 describe the average squared electron-phonon matrix element and M is the average atomic mass. As Ti 2 AlN has higher N(E F ) than the other materials, it should result in a higher T c value if all other effects are similar.

Phonons and electron-phonon interaction.
Phonons have a crucial role in superconductivity, so the calculated phonon dispersion, total and partial vibrational density of states and electron-phonon spectral function for Ti 2 AlX (X: B, C and N) are shown in Fig. 5a-c, respectively. All 3 materials have 8 atoms per primitive unit cell, so there are 3 acoustic and 21 optical phonon modes. A detailed study of the zone centre optical phonon modes is given in our previous paper 21 . As there are no negative frequencies, each structure is dynamically stable. The phonon spectra splits into two distinct frequency regions: a low-frequency region up to 12 THz that contains three acoustic and fifteen optical phonon modes, and a high-frequency region from 15 to 21 THz that contains six optical modes. The E 2g branch of all three materials exhibits a phonon anomaly along the Ŵ-K direction.
There is weak overlap and hybridization of Ti-Al modes in the low-frequency region, and the modes in the high frequency region are dominated by the light X atoms. The DOS in this region has two peaks separated by a small gap for the atoms of Ti 2 AlC and Ti 2 AlN compounds but this gap vanishes for Ti 2 AlB. Overall, the phonon properties of Ti 2 AlX (X: B, C and N) materials are very similar.
The electron-phonon interaction may be studied using the linear response theory 39,40 approach to the Migdal-Eliashberg theory 41,42 . The average electron-phonon coupling constant may be calculated from the Eliashberg spectral function ( α 2 F(ω) ). A detailed study of the Eliashberg spectral function is given in our previous paper 21 . values of Ti 2 AlX (X: B, C and N) have been calculated as 0.685, 0.743 and 0.775, respectively. Figure 5 confirms that (ω) is dominated by the lowest frequency region, in which ∝ ω . The low frequency contribution to the total = is 93%, 97% and 94%, respectively, and is dominated by the coupled motion of Ti and Al atoms. The high-frequency region makes a minor contribution to as this region is dominated by light X atom modes. Using the value of (ω) , the logarithmic average phonon frequency ( ω ln ) is calculated as 219.540 K, 342.218 K and 369.818 K for Ti 2 AlX (X: B, C and N), respectively. The values of and ω ln are used to calculate the superconducting transition temperature T c using the Allen-Dynes modification of the McMillan formula as discussed in our previous paper 21 . In most studies, the value of µ * ranges from 0.10 to 0.16 50,52 . Here, we use µ * =0.13, as this gave the best fit to the experimental T c for Ti 2 GeC, Ti 2 InC, and Ti 2 InN (see Supplementary Information). There is currently no known experimental T c for Ti 2 AlX and so we use µ * = 0.13 and predict T c = 7.8, 10.8, and 13.0 K for Ti 2 AlX (B, C, and N).

Conclusion
We have used a high-throughput approach to study the superconducting properties of 42 different Ti-based M 2 AX phases where A: Al, Si, P, S, Cu, Zn, Ga, Ge, As, Cd, In, Sn, Tl and Pb; X: B, C and N; as shown in Fig. 1 and detailed in Table 2. Currently, 3 are known to be superconducting in experiment (black circles in Fig. 1). Our screening identified that A = Al has the best potential for a high T c , and we then studied the properties of Ti 2 AlX (X:B, C and N) in more detail. The M 2 AX phase with the highest known experimental T c is Nb 2 GeC which has T c = 9.5 K . Our study predicts that Ti 2 AlC will have T c = 10.8 K , the highest T c for a carbide-based M 2 AX phase. Our high-throughput model also predicts the potential for even higher T c in the nitride-based materials, and our detailed calculations predict Ti 2 AlN to have T c = 13.0 K . We also demonstrate superconductivity in the boride-based M 2 AX phases for the first time.
Our analysis shows that the electron-phonon coupling is dominated by low-frequency Ti-based phonon modes, and Ti 3d-based electronic states near the Fermi energy. This work should encourage further studies of superconductivity in M 2 AX phases, and the use of Al rather than the more usual Ge or In should have higher T c and cost savings.
The high-throughput model developed here, with its detailed justification, should also have application in other systematic studies of superconductivity. Figure 5. Phonon dispersion curves, total, partial vibrational density of states and the calculated electronphonon spectral function α 2 F(ω) (red line) and the variation of the electron-phonon coupling parameter (blue line) with rising frequency (ω ) of (a) Ti 2 AlB, (b) Ti 2 AlC and (c) Ti 2 AlN.